Downscaling data assimilation algorithm with applications to statistical solutions of the Navier-Stokes equations

TL;DR

This paper develops a downscaling data assimilation method using a determining map to recover full trajectories from coarse measurements, applied to statistical solutions of the 2D Navier-Stokes equations, with broader applicability to other dissipative systems.

Contribution

It introduces a novel determining map for reconstructing trajectories from coarse data and applies it to statistical solutions of the Navier-Stokes equations, extending the method's generality.

Findings

Statistical solutions are determined by coarse spatial distributions.

The determining map effectively reconstructs full trajectories from coarse data.

The approach is adaptable to other dissipative evolution equations.

Abstract

Based on a previously introduced downscaling data assimilation algorithm, which employs a nudging term to synchronize the coarse mesh spatial scales, we construct a determining map for recovering the full trajectories from their corresponding coarse mesh spatial trajectories, and investigate its properties. This map is then used to develop a downscaling data assimilation scheme for statistical solutions of the two-dimensional Navier-Stokes equations, where the coarse mesh spatial statistics of the system is obtained from discrete spatial measurements. As a corollary, we deduce that statistical solutions for the Navier-Stokes equations are determined by their coarse mesh spatial distributions. Notably, we present our results in the context of the Navier-Stokes equations; however, the tools are general enough to be implemented for other dissipative evolution equations.